CFE-2011, Parallel Sessions, Monday 19/12/2011 Page: 1
Scientific Carbon Stochastic Volatility Model Estimation and Inference:
Forecasting (Un-)Conditional Moments for Options Applications
by
Per Bjarte Solibakkea
a) Department of Economics, Molde University College
Background and Outline
1. The Front December Future Contracts NASDAQ OMX: phase II 2008-2012
No existence of EUAs spot-forward relationship does not exist
EUA options have carbon December futures as underlying instrument
Price dynamics are depending on total emissions
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2. The dynamics of the forward rates are directly specified.
The HJM-approach adopted to modelling forward- and futures prices in commodity markets.
Alternatively, we model only those contracts that are traded, resembling swap and LIBOR models in the interest rate market ( also known as market models). Construct the dynamics of traded contracts matching the observed volatility term structure.
The EUA options market on carbon contract are rather thin, we will therefore estimate the option prices on the future prices themselves. Black-76 / MCMC simulations.
CFE-2011, Parallel Sessions, Monday 19/12/2011
Background and Outline (cont.)
3. Stochastic Model Specification, Estimation, Assessment and Inference
4. Forecasting unconditional Futures and Options Moments,
and measures for risk management and asset allocation
5. Forecasting conditional Futures and Options Moments
i. One-step-ahead Conditional Mean (expectations)
ii. One-step-ahead Standard deviation / Particle filtering
iii. Multi-step-ahead Mean and Volatility Dynamics
iv. Mean / Volatility Persistence
6. Conditional Risk Management and Asset Allocation Measures
7. The EMH case for CARBON commodity markets
Page: 3CFE-2011, Parallel Sessions, Monday 19/12/2011
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The Carbon NASDAQ OMX commodity market
NASDAQ OMX commodities provide access to one of Europe’s leading carbon markets.
350 members from 18 countries covering a wide range of energy producers, consumers and financial institutions.
Members can trade cash-settlement derivatives contracts in the Nordic, German, Dutch and UK power markets with futures, forward, option and CfD contracts up to six years’ duration including contracts for days, weeks, months, quarters and years.
The reference price for the power derivatives is the underlying day-ahead price as published by Nord Pool spot (Nordics), the EEX (Germany), APX ENDEX (the Netherlands), and N2EX (UK).
CFE-2011, Parallel Sessions, Monday 19/12/2011
Indirect Estimation and Inference:
1. Projection: The Score generator (A Statistical Model) establish moments: the Mean (AR-model) the Latent Volatility ((G)ARCH-model) Hermite Polynomials for non-normal distribution features
2. Estimation: The Scientific Model – A Stochastic Volatility Model
where z1t , z2t and (z3t ) are iid Gaussian random variables. The parameter vector is:
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The General Scientific Model methodology (GSM):
2
0 1 1 0 1, 2, 1
1, 0 1 1, 1 0 2
2, 0 1 2, 1 0 3
1 1
22 1 1 1 1 2
22 2 2
3 2 2 1 3 2 1 1 2 3 2 1 1 3
exp( )
1
/ 1 1 / 1
t t t t t
t t t
t t t
t t
t t t
t t t t
y a a y a u
b b b u
c c c u
u z
u s r z r z
u s r z r r r r z r r r r r z
0 1 2 0 1 0 1 1 2 1 2 3, , , , , , , , , , ,a a a b b c c s s r r r
1 10 10 12 2 13 3 1
2 22 2 2
3 33 3 3
exp( )t t t t
t t t
t t t
dU dt U U dW
dU U dt dW
dU U dt dW
SDE:
A vector SDE with two stochastic volatility factors.
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Applications:
Andersen and Lund (1997): Short rate volatilitySolibakke, P.B (2001): SV model for Thinly Traded Equity MarketsChernov and Ghysel (2002): Option pricing under Stochastic VolatilityDai & Singleton (2000) and Ahn et al. (2002): Affine and quadratic term structure modelsAndersen et al. (2002): SV jump diffusions for equity returnsBansal and Zhou (2002): Term structure models with regime-shiftsGallant & Tauchen (2010): Simulated Score Methods and Indirect Inference for Continuous-time Models
3. Re-projection and Post-estimation analysis:
MCMC simulation for Option pricing, Risk Management and Asset allocation Conditional one-step-ahead mean and volatility densities. Forecasting volatility conditional on the past observed data; and/or extracting volatility given the full data series (particle filtering) The conditional volatility function, Multi-step-ahead mean and volatility and mean/volatility persistence. Other extensions.
The General Scientific Model methodology (GSM):
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Stochastic Volatility Models: Simulation-based Inference
Early references are: Kim et al. (1998), Jones (2001), Eraker (2001), Elerian et al. (2001), Roberts & Stamer (2001) and Durham (2003).
A successful approach for diffusion estimation was developed via a novel extension to the Simulated Method of Moments of Duffie & Singleton (1993). Gouriéroux et al. (1993) and Gallant & Tauchen (1996) propose to fit the moments of a discrete-time auxiliary model via simulations from the underlying continuous-time model of interest.
CFE-2011, Parallel Sessions, Monday 19/12/2011
The idea (Bansal et al., 1993, 1995 and Gallant & Lang, 1997; Gallant & Tauchen, 1997):
Use the expectation with respect to the structural model of the score function of an auxiliary model as the vector of moment conditions for GMM estimation.
Replacing the parameters from the auxiliary model with their quasi-maximum likelihood estimates, leaves a random vector of moment conditions that depends only on the parameters of the structural model.
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Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Estimation
Simulated Score Estimation:
Suppose that: is a reduced form model for observed time series, where xt-1 is the state vector of the observable process at time t-1 and yt is the observable process. Fitted by maximum likelihood we get an estimate of the average of the score of the data satisfies:
That is, the first-order condition of the optimization problem.
1 1,
nt t t
y x
Having a structural model (i.e. SV) we wish to estimate, we express the structural model as the transition density , where q is the parameter vector. It can be relatively easy to simulate the structural model and is the basic setup of simulated method of moments (Duffie and Singleton, 1993; Ingram and Lee, 1991).
CFE-2011, Parallel Sessions, Monday 19/12/2011
Details for parameter q estimation:
Compute: where denotes the observed data and n is the sample size. Given a current and the corresponding we obtain the pair as follows (the M-H algorithm):
I. Draw according to
II. Simulate according to
III. Compute and (parameter functionals) from simulation
IV. Define
V. With probability otherwise Page: 9
Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Structural Model Estimation
The scientific model is built using financial market insight/knowledge
Stochastic volatility model computable from a simulation
Metropolis-Hastings algorithm to compute the posterior (only need of a function proportional to the prior)
1,t ty x 0 0 0g
' ',
0 *,q
1 1,
Nt t t
y x
* *g 1 1,
Nt t t
y x
' ' 0 0, , .
Main question: How do the results change as the prior is relaxed?
That is: How does the marginal posterior distribution of a parameter or functional of the statistical model change?
Distance measurement: where Aj is the scaling matrices.
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For a well fitting scientific model:The location measure should not move by a scientifically meaningful amount as k increases. However, the scale measure can increase.
Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Assessment
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Simulated Score Methods and Indirect Inference for Continuous-time Models (some details): Re-projection / Post-Estimation Analysis
Elicit the dynamics of the implied conditional density for observables:
0 1 0 1ˆˆ | ,..., | ,..., ,L L np y y y p y y y
The unconditional expectations can be generated by a simulation:
0ˆ 0 0
ˆ... ,..., ,..., , ...Ln
L L n y yE g g y y p y y d d
ˆN
t t Ly
Let . Theorem 1 of Gallant and Long
(1997) states:
0 1 0 1ˆ ˆ| ,..., | ,..., ,K L K L Kf y y y f y y y
0 1 0 1ˆ ˆlim | ,..., | ,...,K L L
Kf y y y p y y y
We study the dynamics of by using as an approximation. p̂ ˆKf
CFE-2011, Parallel Sessions, Monday 19/12/2011
Application:
Financial CARBON Contracts (EUA)NORD POOL (Phase II: 2008-2012)
Front December Futures Contracts(EUA options will have the December futures as the underlying instrument)
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Objectives (purpose):
Higher Understanding of the Carbon Futures Commodity Markets the Mean equations the Volatility equations
Models derived from scientific considerations and theory is always preferable Fundamentals of Stochastic Volatility Models Likelihood is not observable due to latent variables (volatility) The model is continuous but observed discretely (closing prices)
Bayesian Estimation Approach is credible (densities) Accepts prior information No growth conditions on model output or data Estimates of parameter uncertainty (distributions) is credible
Financial Contracts Characteristics and Risk Assessment & Management The Financial Contracts Characteristics
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Value-at-Risk / Expected Shortfall for Risk Management Stochastic Volatility models are well suited simulation Using Simulation and Extreme Value Theory for VaR-/CVaR-Densities
Simulations and Greek Letters Calculations for Asset Allocation Direct path wise hedge parameter estimates MCMC superior to finite difference, which is biased and time-consuming
Re-projection for Simulations and Forecasting (conditional moments) Conditional Mean and Volatility forecasting Volatility Filtering
The Case against the Efficiency of Future Markets (EMH) Serial correlation in Mean and Volatility Price-Trend-Forecasting models and Risk premiums Predictability
Objectives (purpose): (cont)
CFE-2011, Parallel Sessions, Monday 19/12/2011
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SV models has a simple structure and explain the major stylized facts. Moreover, market volatilities change so frequent that it is appropriate to model the volatility process by a random variable.
Note, that all model estimates are imperfect and we therefore has to interpret volatility as a latent variable (not traded) that can be modelled and predicted through its direct influence on the magnitude of returns.
Mainly three motivational factors:
1. Unpredictable event on day t; proportional to the number of events per day. (Taylor, 86)
2. Time deformation, trading clock runs at a different rate on different days; the clock often represented by transaction/trading volume (Clark, 73).
3. Approximation to diffusion process for a continuous time volatility variable; (Hull & White (1987)
Objectives (why):
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Other motivational factors:
4. A model of futures markets directly, without considering spot prices, usingHJM-type models. A general summary of the modelling approaches for forward curves can be found in Eydeland and Wolyniec (2003).
Matching the volatility term structure.
5. In order to obtain an option pricing formula the futures are modelled directly. Mean and volatility functions deriving prices of futures as portfolios.
Such models can price standardized options in the market. Moreover, the models can provide consistent prices for non-standard options.
6. Enhance market risk management, improve dynamic asset/portfolio pricing, improve market insights and credibility, making a variety of market
forecasts available, and improve scientific model building for commodity markets.
Objectives (why):
CFE-2011, Parallel Sessions, Monday 19/12/2011
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1. NASDAQ OMX Carbon front December contracts
2. The Statistical model and the Stochastic Volatility Model
3. Model assessment (relaxing the prior): model appropriate?
4. Empirical Findings in the mean and latent volatility.
Unconditional mean and latent volatility paths/distributions
Carbon Post-Estimation Analysis:
1. SV-model simulations: Option prices, Risk management and Asset Allocation (unconditional).
2. Conditional mean and volatility, particle filtering, variance functions,
multi-step ahead dynamics and persistence.
5. Conditional Risk Management and Asset Allocation
6. EMH and Model Summary/Conclusion
Carbon Application MCMC estimation/inference:
Assessment
Model Findings
Risk M/Asset Alloc
Conditional Moments
EMH/ Model Summary
Data Characteristics
Estimation Results
Re-projection/Post-Est
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Application Carbon Front December Contracts
Carbon front December Contracts:
Mean / Median / Max. Moment Quantile Quantile K-S RESET Serial dependence
Mode std.dev Min. Kurt/Skew Kurt/Skew Normal Z-test (12;6) Q(12) Q2(12)
-0.04364 0.0000 11.5196 2.84118 0.29749 4.2512 4.59075 70.5138 55.7488 1946.270.00000 2.43729 -10.0083 -0.13418 0.03835 {0.1194} {0.0000} {0.0000} {0.0000} {0.0000}
BDS-statistic (e=1) KPSS (Stationary) Augmented ARCH VaR CVaRm=2 m=3 m=4 m=5 Level Trend DF-test (12) 2.5/0.5% 2.5/0.5%
16.6788 23.5820 30.1427 38.4401 0.14330 0.14340 -56.0675 594.675 -5.247 -7.178{0.0000} {0.0000} {0.0000} {0.0000} {0.4121} {0.0568} {0.0000} {0.0000} -8.311 -9.694
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Scientific Models: Stochastic Volatility Model /Parameters (q)
Bayesian Estimation Results1. Several serial Bayesian runs establishing the mode
We tune the scientific model until the posterior quits climbing and it looks like the mode has been reached:
2. A final parallel run with 24 (8 *3) CPUs and 240.000 MCMC simulations(OPEN_MPI (Indiana University) parallell computing)
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Application Carbon Front December Contracts
Carbon Front December General Scientific Model. Statistical Model SNP-11116000 - fit modelParameter values Scientific Model. Standard Parameters Non-linear-GARCH. Standard Mode Mean error Mode Mean error
a0 0.026974 0.033957 0.041845 n1 a0[1] 0.010997 0.017017 0.012873
a1 0.053948 0.045583 0.021425 n2 a0[3] 0.009816 -0.027176 0.015376
b0 0.630520 0.624160 0.078653 n3 a0[4] -0.007590 -0.005462 0.003885
b1 0.985140 0.947710 0.038068 n4 a0[5] 0.071771 0.104291 0.017859
c1 0.577490 0.663590 0.080555 n5 a0[6] 0.001586 0.002598 0.003412
s1 0.062399 0.068591 0.016147 n6 A(1,1) 0.004190 -0.000290 0.005238
s2 0.226330 0.196810 0.032872
r1 -0.432440 -0.385280 0.113010 n7 B(1,1) 0.072114 0.043127 0.046907n8 R0[1] 0.151411 0.265661 0.062968
log sci_mod_prior 5.797190 n9 P(1,1) 0.326412 0.430157 0.089448
log stat_mod_prior 0.000000 c2(3) = n10 Q(1,1) 0.926579 0.860786 0.041011
log stat_mod_likelihood -1515.8624 -0.94841 n11 V(1,1) -0.116156 0.037407 0.139785log sci_mod_posterior -1510.0652 {0.81373}
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Scientific Model: Model Assessment – the model concert test
Carbon front December k = 1, 10, 20 and 100 densities – reported.
Application Carbon Front December Contracts
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Scientific Model: The Stochastic Volatility Model: log-sci-mod-posterior
Log sci-mod-posterior (every 25th observation reported): Optimum is along this path!
Application Carbon Front December Contracts
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Scientific Model: Carbon q-paths and densities; 240.000 simulations
Application Carbon Front December Contracts
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Scientific Model: Stochastic Volatility
The chains look good. Rejection rates are:
The MCMC chain has found its mode.
A well fitted scientific SV model: The result indicates that the model fits and that location measure is stable and the scale measure increases, indicating that the scientific model has empirical content.
Reported Proportion Number of Proportion%-rejected Moved Rejects Accepted
theta1 ( 1) 0.49051424 0.1255875 60.525 125.5875
theta2 ( 2) 0.47925517 0.1248125 59.7375 124.8125
theta3 ( 3) 0.47381869 0.12480417 59.1625 124.804167
theta4 ( 4) 0.4807526 0.12526667 60.2333333 125.266667
theta5 ( 5) 0.47864768 0.1262625 60.4583333 126.2625
theta6 ( 6) 0.47833745 0.12455 59.5333333 124.55
theta7 ( 7) 0.48576032 0.12464583 60.5958333 124.645833
theta8 ( 8) 0.48436667 0.12407083 64.1208333 124.070833
Sum 0.48436667 1 484.366667 1000
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Empirical Model Findings:
For the mean stochastic equation: Positive mean drift (a0 = 0.026; s.e. = 0.03) and serial correlation (a1 = 0.054;
s.e. 0.021) for the CARBON contracts
For the latent volatility: two stochastic volatility equations: Positive constant parameter (e0.6305 >> 1) Two volatility factors (s1 = 0.0624, s.e.=0.0161; s2 = 0.2263, s.e.=0.0329)
Persistence is high for s1 with associated (b1 = 0.985, s.e. = 0.0381) ; persistence is lower for s2 with associated (b2 = 0.5775, s.e.=0.0806)
Asymmetry is strong and negative (r1 = -0.4324, s.e.=0.1130)
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Scientific Model: The Stochastic Volatility Model.
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The Option market versus SV-model prices 05.09.2011
Application Carbon Front December Contracts
Option Prices 05.09.2011 Market closing prices SV-Model pricesDEC-11 Strike Price call Dec-11 put Dec-11 call Dec-11 put Dec-11 Volume 0 6.0 2.59 0.06 2.54 0.04
0 6.5 2.15 0.12 2.09 0.080 7.0 1.72 0.19 1.66 0.130 7.5 1.33 0.29 1.30 0.230 8.0 0.97 0.43 0.92 0.390 8.5 0.67 0.62 0.62 0.610 9.0 0.46 0.91 0.41 0.890 9.5 0.31 1.25 0.26 1.230 10.0 0.2 1.64 0.17 1.620 10.5 0.12 2.06 0.11 2.110 11.0 0.07 2.51 0.06 2.520 11.5 0.04 2.97 0.04 3.040 12.0 0.03 3.45 0.03 3.510 12.5 0.01 3.93 0.01 3.960 13.0 0.01 4.42 0.01 4.47
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0
Option
Prices
Strike Prices
Carbon Option Market and SV-Model prices Maturity 2011 for 2011/09/05
Market closing prices call Dec-11 put Dec-11 SV-Model prices call Dec-11 put Dec-11
Scientific Model: The Stochastic Volatility Model.
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Risk assessment and management: CARBON VaR / CVaR
Application Carbon Front December Contracts
Scientific Model: The Stochastic Volatility Model.
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Asset Allocation/Dynamic Hedging: CARBON GREEK Letters
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Scientific Model: Re-projections / nonlinear Kalman filtering
Of immediate interest of eliciting the dynamics of observables:
0 1 0 0 1 0( | ) ( | , )k ky x y f y x dy One-step ahead conditional mean:
One-step ahead conditional volatility:
'0 1 0 0 1 0 0 1 0 1 0( | ) ( | ) ( | ) ( | , )k kVar y x y y x y y x f y x dy
Filtered volatility is the one-step ahead conditional standard deviation evaluated at data values:
where yt denotes the data and yk0 denotes the kth element of the vector y0, k = 1,…M.
For instance, one might wish to obtain an estimate of:
for the purpose of pricing an option (from the re-projection step).
1 10 1 ,..., )( | ) |t L tk x y yVar y x
Application Carbon Front December Contracts
*1 2exp( )
t T
t
t
v v dt
CFE-2011, Parallel Sessions, Monday 19/12/2011
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SV Model: One-step-ahead conditional moments
0 1 0 0 1 0( | ) ( | , )k ky x y f y x dy '0 1 0 0 1 0 0 1 0 1 0( | ) ( | ) ( | ) ( | , )k kVar y x y y x y y x f y x dy
Application Carbon Front December Contracts
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SV Model: filtered volatility /particle filtering for Option pricing
1 10 1 ( ,..., )| | 0,...,t L tk x y yy x t n
Application Carbon Front December Contracts
0
0.05
0.1
0.15
0.2
Conditonal
Mean
Density
One-step-ahead density fK(yt|xt-1,) xt-1 =-10,-5, -3, -1, m, 0, +1, +3, +5, +10%
Frequency xt-1=-10% Frequency xt-1=-5% Frequency xt-1=-3% Frequency xt-1= Mean (-0.032)
Frequency xt-1=0% Frequency xt-1=+3% Frequency x-1=+5% Frequency x-1=+10%
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
GAUSS-hermite Quadrature Density Distribution
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SV Model: Conditional variance functions (asymmetry)
(shocks to a system that comes as a surprise to the economic agents)
Application Carbon Front December Contracts
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SV Model: Multistep-ahead volatility dynamics
(volatility impulse-response profiles)
0
0.5
1
1.5
2
2.5
3
Var
ian
ce E
[Var
(yk
,j|x
-1)
DAYS
Multistep Ahead Dynamics s2j
dy0 dy-1 (low) dy+1 (high) dy-3 (low) dy+3 (high) dy-6 (low) dy+6 (high) dy-10 (low) dy+10 (high)
Application Carbon Front December Contracts
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SV Model: Mean and Volatility Persistence (half-lives = –ln2 / b)
Application Carbon Front December Contracts
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Mean
Days
CARBON Profile Bundles for the MEAN (overplots of profiles)
0
3
6
9
12
15
18
21
24
27
30
Vol
atil
ity
Days
CARBON Profile Bundles for the VOLATILITY (overplots of profiles)
Halflives:
28.238149SE=1.324
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Scientific Model Re-projections: Conditional SV-model moments:
Conditional VaR/CVaR for RM and Greeks for Asset allocation
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Application Carbon Front December Contracts
CFE-2011, Parallel Sessions, Monday 19/12/2011
Scientific Model Reprojections: Extensions using SV-model simulations:
Realized Volatility and continuous / jump volatility (5 minutes simulations):
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Application Carbon Front December Contracts
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0.00016
0.00018
0.0002
Realized Volatility
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0.00016
0.00018
0.0002Continuous Volatility
-0.00003
-0.00002
-0.00001
0
0.00001
0.00002
0.00003
Jump Volatility
CFE-2011, Parallel Sessions, Monday 19/12/2011
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Scientific Model Re-projections: Post Estimation Analysis
Post estimation analysis add new information to market participants:
Option prices for any derivative for any maturity. Credible densities are available for all call/put prices.
Credible densities for VaR/CVaR and Greek letters are available for risk management and asset allocation
Conditional mean (expectations) is narrow information from the history?
The filtered volatility (particle filter) add information for the one-day-ahead conditional volatility. Conditional return densities for obs. Xt-1. Gauss quadrature densities are available.
Conditional variance functions evaluates the surprise to economic agents from market shocks.
Multi-step-ahead dynamics for the mean and volatility are available
Conditional Risk management and asset allocation measures available
Realized Volatility can be obtained from simulation step change (96 steps per day = 5 minutes data). Page: 43
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CARBON front December contracts and EMH:
Drift in the mean (risk premium) is positive but negligible (insignificant)
The positive serial correlation in the mean (0.054) is probable not tradable
The volatility clustering is strong (0.985) but probably not tradable
Asymmetry is strong (-0.432) but not tradable
The mean and volatility is stochastic and not predictable
EMH (weak form/semi-strong form) seems clearly acceptable.
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Application Carbon Front December Contracts
CFE-2011, Parallel Sessions, Monday 19/12/2011
Main Findings for CARBON front December contracts: Stochastic Volatility models give a deeper insight of price processes and
the stochastic flow of information interpretation
The Stochastic Volatility model and the statistical model seem to work well in concert (indirect estimation)
The MC chains look good and rejection is acceptable giving a reliable and viable stochastic volatility model
The SV-model results induce serial correlation in mean and volatility, persistence and negative asymmetry. One volatility factor is slowly moving while the second is quite choppy.
Option Prices can easily be generated for any maturity. We compared two maturities market prices to model prices (mean and distributions).
Risk management procedures are available from Stochastic Volatility models and Extreme Value Theory (VaR/CVaR and Greek letters)
Conditional moments, particle filtering and volatility variance functionsinterpret asymmetry, pricing options and evaluates shocks.
Imperfect tracking (incomplete markets) suggest that simulation is a well-suited methodology for derivative pricing Page: 46
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